Inflation, phase transitions and the cosmological constant
aa r X i v : . [ g r- q c ] F e b Inflation, phase transitions and the cosmologicalconstant
Orfeu Bertolami ∗ Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias, Universidade do PortoRua do Campo Alegre, 4169-007 Porto, Portugal
Abstract
Cosmological phase transitions are thought to have taken place at the early Universeimprinting their properties on the observable Universe. There is strong evidence that,through the dynamics of a scalar field that lead a second order phase transition, infla-tion shaped the Universe accounting for the most conspicuous features of the observedUniverse. It is shown that inflation has also striking implications for the vacuum energy.Considerations for subsequent second order phase transitions are also discussed. ∗ Also at Centro de F´ısica das Universidades do Minho e do Porto, Rua do Campo Alegre,4169-007 Porto, PortugalE-mail address: [email protected] 1
Introduction
Inflation endows the Universe with its most salient observational features. One of the firstrealisations of inflation involved a first order cosmological phase transition driven by the Higgsfield of a Grand Unified Theory (GUT) where bubbles of the new vacuum nucleated at the oldvacuum of the GUT. The completion of this scenario required a successful expansion of thenew vacuum bubbles, their collision and an efficient percolation so to homogeneously fill andreplace entirely the old vacuum [1]. However, it was soon understood that inflation driven bythe Higgs field of a GUT could not warrant all these conditions and, furthermore, it did notallow for quantum field fluctuations to generate the needed energy density fluctuations to formstructures. Therefore, models with a potential that allowed for a second order phase transitionwere since then favoured [2, 3] (see e.g. Ref. [4] for a review). After inflation the Universe issupercooled and a suitable mechanism for its reheating must follow in order to lead to a hotUniverse compatible with nucleosynthesis, the cosmic microwave background radiation, etc.In what concerns the vacuum energy, as pointed out by Zeldovich [9] long ago, that itgravitates and Lorentz invariance implies that it must be included into Einstein’s field equationsas cosmological constant, that is, its energy-momentum tensor has the form T ab = Λ8 πG g ab .Thus, the associated vacuum energy, ρ V , corresponds to the T component while the vacuumisotropic pressure, p V , to the T ii components. Explicit computations of the zero-point energyin the momentum space in the context of field theory (see, for instance, Ref. [8]) imply thatthese components must satisfy the equation of state: p V = − ρ V . (1)Field theory estimates lead to the well known conclusion that a severe fine tuning is needed intothe Einstein’s field equations to match the observed value: Ω Λ ≃ .
7, or ρ V ≃ . × − GeV for h = 0 .
7, which compares with the contribution from the Standard Model (SM) afterthe Higgs field acquires a non-vanishing vacuum expectation value, ρ ( SM ) V = O (250 GeV ) , adiscrepancy of a factor 10 [5, 6, 7]. GUTs and quantum gravity lead to even more dramaticdiscrepancies. This is the well known cosmological constant problem (see, for instance, Refs.[10, 11, 12, 13] for reviews and discussions).In what follows we shall argue that generic thermodynamic considerations for the vacuumtogether with the well known Bekenstein bound for entropy [14] in the context of the newinflationary scenario imply in an exponential suppression of the vacuum energy once the in-flationary phase transition is completed. Further arguments will be presented to support thatsimilar considerations might apply to subsequent second order cosmological phase transitions. Let us now focus on the transition that took place during the inflationary process where the oldor “false” vacuum was dynamically driven into the new “true” vacuum. The matter sector ofthe Universe was then dominated by a generic inflaton field, initially at the top of its potential2t a thermal bath at temperature T ( i ) . The starting point to describe the vacuum evolutionprocess is the Gibbs-Duhem equation: SdT − V dp + n X j =1 N j dµ j = 0 , (2)where, S , N j and µ j ( j = 1 , ..., n ) are the entropy, the number of particles and the chemicalpotential of each species, respectively . These quantities refer to the vacuum, but in order notto overburden the notation no explicit notation will be introduced to distinguish their nature;however, we must discriminate, following Coleman and co-workers [15], the ”true” final vacuumfrom the initial ”false” vacuum. For the two vacua, which presumably have no free particles, N j = 0. It then follows a well known relationship, which must be implemented with a minussign in order to account for the negative nature of the vacuum pressure: dpdT = − ∆ S ∆ V ≃ − SV , (3)where the approximation is due to inflation as V ≫ V i , where V i is the initial volume, and asfrom the Third Law of Thermodynamics it is expected that S i ≃ S ≤ πkRE ~ c , (4)where k is Boltzmann’s constant, R is the system length scale and E its energy. We saturatethis bound considering for the length scale the horizon distance; hence Eq. (3) can be writtenas: dpdT ≃ − πkR Hor E ~ cV = − πkR Hor ρ ~ c . (5)However, using Eq. (1), we get after integration: p T p F ≃ exp (cid:18) πkR Hor ∆ T ~ c (cid:19) ≃ exp − πR Hor E ( i ) T h ~ c ! , (6)where we have introduced the thermal energy, E ( i ) T h , at the onset of inflation and used that dueto the supercooling ensued by inflation that T f → R Hor > e R ( i ) Hor ≃ e M − P and E ( i ) T h ≃ − M P , where M P is Planck’s mass. Hence, writing the result in terms of the falsevacuum energy, ρ F ≃ M p : ρ T < e − πe M P = 10 − . × M P , (7)a quite considerable suppression. This result has at least two relevant implications. The obviousfirst one is that no fine tuning is now required. The second one it that dark energy, assumedto be responsible for the late time accelerated expansion of the Universe, is not due to anyresidual left by the quantum gravity vacuum or the GUT vacuum.3 Subsequent phase transitions
The mechanism described above can account for the suppression of the cosmological constantafter inflation, however the cosmological constant may pick other contributions, for instance,from the electroweak phase transition and the quark-gluon phase transition. It is of courseimpossible to apply the above considerations to subsequent phase transitions and furthermore,specific knowledge of these transitions should be known. However, some generic features ofthe mechanism outlined above can considered as a suitable guideline. In fact, an exponentialchange of the pressure arising from Eq. (3) can be found, for example, in the Clausius-Clapeyronequation for the change of the critical points with pressure and in the Kelvin-Helmholtz equationfor the pressure change due to capillaries in fluids. In the Clausius-Clapeyron equation, theargument of the exponential contains essentially the ratio between the latent heat to the thermalenergy. In the Kelvin-Helmholtz equation, the argument of the exponential involves the ratioof the surface tension times the curvature of the meniscus to the thermal energy, and a pressuresuppression arises when the meniscus is concave. A generic way to express the main featuresof these equations would be: ln (cid:18) p T p F (cid:19) = − V eff E T h , (8)where V eff is a generic energy difference that effectively plays the role of a potential. A signif-icant suppression might ensue whether V eff /E T h ≫
1. If this behaviour can be encountered incosmological phase transitions after inflation, a suppression of the vacuum energy may ensue.Unfortunately, it seems very difficult to advance with general conclusions without specificdetails of the phase transition in question. However, a conspicuous property of the phase tran-sition implied by the new inflationary type models is that it involves a spinodal decomposition ,in opposition to the nucleation process that characterises the first order phase transitions. Thegeneric properties of the second order phase transitions are described by the Cahn-Hilliardmodel [16]. In this model the free energy is given in terms of the gradient of the concentration, c ( x ), of each phase, where x is a typical length scale between the phases. For a two domainssystem, the two phases correspond to c = ±
1. The free energy variation is given by [16]:∆ F = Z γ |∇ c | dV , (9)where √ γ sets the length scale of the transition regions between the domains. Using p = − (cid:18) ∂F∂V (cid:19) T , (10)in order to relate with the vacuum pressure and assuming the equilibrium solution of thediffusion equation of the model, the Cahn-Hilliard equation, c ( x ) = tanh ( x/ √ γ ), hence for x ≫ p ≃ p → x → ∞ ) and therefore, the vacuum energy tends to vanish.Of course, these considerations are very general and may not hold under a more detailedanalysis, however they do not seem to be ruled out neither by theoretical considerations [17,18, 19] nor by some specific numerical simulations [20, 21, 22].4 Discussion and Conclusions
The vacuum is an elusive concept. From just a featureless ground state in classical field theoryand quantum mechanics it acquires a rich structure in quantum field theory. The pioneeringefforts of Dirac, Zeldovich, Hawking, Coleman, Kibble and others have shown that the vacuumgravitates, it can give origin to a thermal radiation, undergo to phase transitions and can giverise to topological defects likewise in material systems. The quantum vacuum can also be at thecore of the spontaneous breaking of symmetries, a crucial feature of the Standard Model of Fun-damental Interactions of Nature. However, this endowed protagonism poses an embarrassingproblem of fine tuning, the cosmological constant problem, whose solution has been the objectof countless attempts from a wide range of points of view. Given the need to bring gravitywithin the context of a quantum framework, it is often argued that the cosmological constantproblem cannot be properly addressed without a suitable fundamental quantum gravity the-ory. It might be very well the case, however attempts in the context of, for instance, stringtheory, one of the most accomplished proposals to quantise gravity and unify all interactionsof Nature, has not shown to be successful in this respect [23], despite the interesting ideas thatspring from the vacua landscape of string theory [24, 25], from M-theory compactifications [26]and other multiverse considerations [27]. The same can be stated about other proposals toquantise gravity such as loop quantum gravity, although recent hopeful and interesting results[28].Nevertheless, our proposal is much more modest. We argue that the universal laws of ther-modynamics should also apply to the vacuum, as already considered for black holes and inattempts to endow the gravitational field with entropy [29], and furthermore, that inflationprovides a suitable way out for the fine tune problem of the the vacuum energy. Thus, asthe inflaton evolves and drives the inflationary accelerated expansion that shapes most of theobservable properties of the Universe, it also induces thermodynamical transformations in thevacuum that yield in its suppression. In fact, the idea that the vacuum evolves [30, 31, 32] ina smooth way with the cosmic time, t , actually as Λ ∼ t − , allows for interpolating, in specificcontexts, the expected O ( M P ) initial value of the cosmological constant to its observationalvalue. Thus, if the vacuum evolves, it is not all that surprising that an abrupt and significantevent such as inflation can have an even a more dramatic impact on the vacuum energy. Thearguments presented in section 2 seem to support this conjecture. It remains to be seen whetherthe presented assumptions can also be used for generic cosmological phase transitions, as pro-posed in section 3; if so, the several avatares of the cosmological constant can be all tackledwith the same underlying set of arguments.In conclusion we should stress that the paradigmatic nature of inflation is reinforced. Indeed,besides the well known virtues of solving the initial conditions problems of the Big Bang model,of explaining the absence of topological defects such as magnetic monopoles and of giving riseto the energy density fluctuations responsible for structure formation, the arguments presentedin this work indicate that the accelerated expansion provided by inflation induce changes inthe vacuum that suppress its energy density. Thus, given that inflation is such a fundamentalingredient for solving so many issues, the question of the initial conditions that allow for itsonset might hold the key to discriminate the viability scenarios arising from the very early5niverse quantum gravity epoch. References [1] A.H. Guth, Phys. Rev.
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